For the strong nonlinear case, there emerges an energy contour line which is just a horizontal line, as can be seen in Fig. REF (c) and (d). This line corresponds to \(\xi =-T^+/T^-\)  with the corresponding \(\theta \) -independent energy \(E=H_c(\xi =-T^+/T^-,\theta )=c(T^+/T^-)^2\) . However, for this same energy, one can find another energy contour line determined by \(H_c(\xi , \theta )=E\)  or \(c(\xi -\frac{T^+}{T^-})+2NT^-\sqrt{1-\xi ^2}\cos \theta =0\) . This is represented by the wavy thick black line in Fig. REF (c) and (d). As \(|T^+/T^-|\)  decreases, these two contour lines eventually intersect with each other. This is illustrated in Fig. REF (d), and in Fig. REF (e) we isolated out these two lines. The intersections of these two lines represent the saddle points of the system, and these two intersecting lines thus become the separatrix of the system [1]}. The direction of the motion along the separatrix are marked by red arrows in Fig. REF (e). The trajectory of the pendulum length when the system is prepared on the separatrix is depicted in Fig. REF (h). It looks like the combination of the STO and the OTT trajectories plotted in Fig. REF (f) and (g). In real situation, however, the system can come very close to, but never touch, the separatrix. In the region enclosed by the two separatrix, the STO mode arises naturally.
